WeierstraB-Institut fiir Angewandte Analysis und Stochastik im Forschungsverbund Berlin e.V. The low-temperature phase of Kac-Ising models Anton Bovier1 , Milos Zahradnik2 submitted: 8th May 1996 1 WeierstraB-Institut 2 Department of Mathematics fiir Angewandte Analysis Charles University und Stochastik Sokolovska 83 MohrenstraBe 39 186 00 Praha 8 D - 10117 Berlin Czech Republic Germany e-mail: [email protected] e-mail: [email protected] Preprint No. 240 Berlin 1996 Key words and phrases. Ising models, Kac potentials, low temperature Gibbs states, contours, Peierls argument. Work partially supported by: Commission of the European Union under contracts CHRX-CT93- 0411 and CIPA-CT92-4016, Czech Republic grants c. 202/96/0731 and c. 96/272. Edited by WeierstraB-Institut fiir Angewandte Analysis und Stochastik (WIAS) MohrenstraBe 39 D -- 10117 Berlin Germany Fax: + 49 30 2044975 e-mail (X.400): c=de;a=d400-gw;p=WIAS-BERLIN;s:._preprint e-mail (Internet): [email protected] Abstract: We analyse the low temperature phase offerromagnetic Kac-Ising models in dimensions d ~ 2. We show that if the range of interactions is ,-1, then two disjoint translation invariant Gibbs states exist, if the inverse temperature /3 satisfies f3 - 1 ~ 'Yf;,, where "' = (2d~~)(~~l), for any € > 0. The prove involves the blocking procedure usual for Kac models and also a contour representation for the resulting long-range (almost) continuous spin system which is suitable for the use of a variant of the Peierls argument. 1. Introduction In 1963 Kac et al. [KUH] introduced a statistical mechanics model of particles interacting via = long, but finite range interactions, i.e. through potentials of the form J (r) 1dJ(rr), there J is 1 some function of bounded support or rapid decrease (the original example was J(r) = e-r) and 'Y is a small parameter. These models were introduced as microscopic models for the van der Waals theory of the liquid-gas transition. In fact, in the context of these models it proved possible to derive in a mathematically rigorous way the van der Waals theory including the Maxwell constructi<?n in the limit 'Y-!. 0. In mathematical terms, this is stated as the Lebowitz-Penrose theorem[LP]: The distribution of the density satisfies in the infinite volume limit a large deviation principle with a rate function that, in the limit as 'Y tends to zero, converges to the convex hull of the van der Waals free energy. For a review of these results, see e.g. the textbook by Thompson [T]. Only rather recently there has been a more intense interest in the study of Kac models that went beyond the study of the global thermodynamic potentials in the Lebowitz-Penrose limit, but that also considers the distribution of local mesoscopic observables. This program has been carried out very nicely in the case of the Kac-Ising model in one spatial dimension by Cassandra, Orlandi, and Presutti [COP]. A closely related analysis had been performed earlier by Bolthausen and Schmock [BS]. These analysis can be seen as a rigorous derivation of a Ginzburg-Landau type field theory for these models. Very recently, such an analysis was also carried out in a disordered version of the Kac Ising model, the so-called Kac-Hopfield model by Bovier, Gayrard, and Picco [BGP1,BGP2]. An extension of this work to higher dimensional situations would of course be greatly desirable. This turns out to be not trivial and, surprisingly, even very elementary questions about the Kac model in d ~ 2 are unsolved.. One of them is the natural conjecture that the critical inverse tem- perature f3c( 1) in the Kac model should converge, as 'Y -!-0, to the mean-field critical temperature. This conjecture can be found e.g. in a recent paper by Cassandra, Marra, and Presutti [CMP]. In that paper a lower bound f3c (r) ~ 1 + lry2 I ln 11 is proven. A corresponding upper bound is only known in a very particular case where reflection positivity can be used [BFSJ. 1 In addressing this question one soon finds the reason for this unfortunate state of affairs. All the powerful modern methods for analyzing the low-temperature phases of statistical mechanics models, like low-temperature expansions and the Pirogov-Sinai theory, have been devised in view of models with short range (often nearest neighbor) interactions, with possible longer range parts treated as some nuisance that can be shown to be quite irrelevant. To deal with the genuinely long-range interaction in Kac models, that is to exploit their long range nature, these methods require substantial adaptation. The purpose of the present paper is to help to develop adequate techniques to deal with this problem - that beyond proving the conjecture of [CMP] will, hopefully, also provide a basis for the analysis of disordered Kac models. (Together with possible other .means not touched by the presented article : most notably with suitably developed expansion techniques for long range models). The model we consider is defined as follows. We consider a measure space (S, :F) where = S {-1, l}zd is equipped with the product topology of the discrete topology on {-1, 1} and :Fis the corresponding finitely generated sigma-algebra. We denote an element of S by O" and call it a spin-configuration. If A C ::zd, we denote by O" A the restriction of O" to A. For any finite voluI]le A we define the energy of the configuration O" A (given the external configuration O" Ac) as (1.1) iEA,j<f.A = f where J-y(i) l'dJ('yi) and J: IRd--+ IR is a function that satisfies m,d dxJ(x) = 1. For simplicity we will assume that J has bounded support, but the extension of our proof to more moderate assumptions on the decay properties of J is apparently not too difficult. To be completely specific = we will even choose J(r) cdlilxl:::;l where cd normalizes the integral of J to one1; Here I· I is most conveniently chosen as the sup-norm on IRd. Finite volume Gibbs measures ("local specifications") are defined as usual as (1.2) where z~,/3,A is the usual partition function. Note that under our assumptions on J the local specifications for given A depend only on finitely many coordinates of 'T/· An infinite volume Gibbs state µ-y,/3 is a probability measure on (S, :F) that satisfies the DLR-equations (1.3) Our first result will be the following 1 The generic name Cd will be used in the sequel for various finite, positive constants that only depend on di~ension. 2 Theorem 1: Let d ~ 2. Then there exists a function f (r) with lim.y.1-o f (r) = 0 such that for all f3 > 1 + f (/ ), there exist at least two disjoint extremal infinite volume Gibbs states with 1-E local specifications given by {1.2). Moreover, for 'Y small enough, f(r) ::; 1<2d+2><1+1/d) for arbitrary E>O Remark: This theoreI? shows that the conjecture of [CMP] is correct. Together with Theorem 1 = of [CMP] it implies that lim.y.1-0 f3c(r) 1 in the Kac model. While completing this work we have received a paper by M. Cassandra and E. Presutti [CP] in which the conjecture of [CMP] is also proven, but no explicit estimate on the asymptotics of the function f (' Y) is given. Their proof is rather different from ours. Although at the moment we make use of the spin flip symmetry of the model, the contour language we introduce is also intended as a preparatory step for future use of the Pirogov-Sinai theory for non-symmetric long range models. We will in fact get more precise information on the infinite volume Gibbs measures in the course of the proof. This will be expressed in terms of the distribution of "local magnetization", mx(CT), defined on some suitable length scale 1 « f, « ,-1. Given such scale l, we will partition the lattice 7Ld into blocks, denoted by x of side length l. We set Identifying the block x with its label x E 7L, we could thus set = x {i E 7Ld I Ii - lxl ::; l/2} (1.4) We then define 1 m (CT) = - "'CT· (1.5) x - f, ~ '£ iEx In the sequel we will assume that all finite volumes we consider are compatible with these blocks, that is are decomposable into them. We will also assume that 1l is an integer. For any volume A compatible with the block structure, we denote by MA C :FA the sigma-algebra generated by the family of yariables {mx(CT)}xEA· The block variables will be instrumental in the proof of Theorem 1. However, they are also the natural variables to characterize the nature of typical configurations w.r.t. the Gibbs measure. We should note that this first step of passing to the variables mx(CT) is also used in [CP], in fact it is used in virtually all work on the Kac model. The remainder of this article is organized as follows. In Section 2 the distribution of the block spins are formally introduced and the block-spin approximation of the Hamiltonian is discussed. In Section 3 we introduce our notion of Peierls contours and prove our theorem through variant of the Peierls argument [P]. Acknowledgements: We thank Errico Presutti and Marcio Cassandro for sending us a copy of their paper [CP] prior to publication. M. Zahradnik also acknowledges useful discussions with E. Presutti on Kac models in general and about their recent preprint in particular. We would like to 3 thank also the home institutions of the authors and the Erwin Schrodinger Institute in Vienna for hospitality that made this collaboration possible. 2. Block spin approximation All the questions we want to answer in our model will after all concern the probabilities of events that are elements of the sigma-algebras M v for finite volumes V. If A E M v is such an event and A ~ V, we have the following useful identity (2.1) rn:i:,:i:EV {rn:i: }CA The sum over mx runs of course over the values { -1, -1 + u-d, ... , 1-2.e,-d, 1} Note that we ]Ilay, if J has compact support, assume without loss of generality that A is sufficiently large so that the local specification µu11.f3\vv,TJ11.c does not depend on 'T/· We will therefore drop the 'TJ in this expression. 'Y' ' The main point which makes the Kac-model special, is that the Hamiltonian is "close" to a function of the block spins. Namely, we may write x,yEV iEx,jEy xEV,yEVc iEx,jEy x,yEV iE:z:,jEy (2.2) x,yEV iE:z:,jEy = H~~1,v(mv(O"v ), mvc (O"vc )) + AH-y,t.,v(O"v, O"yc) where we have set (recall that J'Y(.f.x) = .e,-dJt.'Y(x)) x,yEV :z:EV,yEYc and L L AH-y,t.,V(O"v, O"yc) = -~ [J-y(i - j) - J'Y(.f.(x - y))] O"iCJj x,yEV iEx,jEy (2.4) L L [J'Y(i - j) - J'Y(.e.(x - y))J O"iO"j xEV,yEVc iE:z:,jEy 4 Lemma 2.1: For any V c zzd, suplAH ,t,v(crv,crvc)I:::; cdrllVI (2.5) 7 (j where cd is some numerical constant that depends only on the dimension d. Proof: This fact is well-known and simple for all Kac models. In our case it follows from the observation that [1 (i - j) - 1 (£(x -y))] = 0, unless Ix - yj ~ l/(1l).<) 7 7 As consequence of Lemma 2.1 we get the following useful upper and lower bounds for the distribution of the block spins: -/3ldH(o~v(mv,.mvc)IT IE 1f µ -Oy", J/3\\ 'v v ( mv ) >< ~. e -f3£,d, HI <o> (mv,mvc): zI:TEV OI" Em :1zf: (u)=m:z: e ±/3cd-yljVI (2.6) 6mv e -y,l,V xEV o- m:i:(o-)=m:i: Of course if f.d /mx/2 E zt (2.7) else and thus, by Sterling's formula, (2.8) where I(m), form E [-1, 1] is l+m 1-m I(m) = - -ln(l + m) + - -ln(l- m) (2.9) 2 2 Therefore we define L L L E-y,/3,£,v(mv,mvc) = -~ 1 t(x-y)mxmy- 1-yt(x-y)mxmy+(3-1 I(mx) (2.10) 7 x,yEV xEV,yEVc xEV to get Lemma 2.2: For any finite volume V and any configuration mv, we have < e-/3ld E1,/3,l, v (mv,mvc (o-vc )) o-A\v (m ) e±/3cn£1VI (2.11) µ-y,/3,V V > ~ e_,f3£dE ,13,e,v(mv,mvc(uvc)) 6mv 1 Remark: £will be chosen as tending to infinity as')' tends to zero. The idea is that that E ,/3,l,V is 7 in a sense a "rate function"; that is to say, E-y,/3,l,V alone determines the measure since the residual entropy is only of the order di~l IVI· The problem is that this is only meaningful when we consider events A for which the minimal E-y,/3,l,V is of order !VI above the ground state to make sure that 5 neither the residual entropy nor the error terms in (2.11) may invalidate the result. We will have to work in the next section to define such events. It is instructive to rewrite the functional E'Y,/3,l,V in a slightly different form using that -mxmy = ~(mx - my)2 - Hm; + m~) (we drop the indices (, (3, f, henceforth but keep this dependence in mind). We set (2.12) where f 13 is the well-known free energy function of the Curie-Weiss model, (2.13) Then Ev(mv,mvc) = Ev(mv,mvc) - Cv(mvc) (2.14) where (2.15) vc. depends only on the variables on The form· Ev makes nicely evident the fact that the energy functional favours configurations that are constant and close to the minima of the Curie-Weiss function f /3 ( m). 3. Peierls contours In this Section we define an appropriate notion of Peierls-contours in our model and use this to proof Theorem 1 by a version of the Peierls argument2• The general spirit behind the definition of Peierls contours can be loosely characterized as follows: We want to define a family of local events that have the property that at least one of them has to occur, if the effect of boundary conditions does not propagate to the interior of the system. Then one must show that the probability that any of these events occurs is small. We will define such events in terms of the block spin variables m~ (a). More precisely, since it is crucial for us to exploit that the new interaction is still long 2 While the proof of [CP] is also based on a Peierls argument, their definition of Peierls contours is completely different from ours. 6 range3, contours will be defined in terms of the local averages, c/Jx ( m), and the local variances, 'l/Jx(m), defined through = L ef>x(m) J 1,(x - y)my (3.1) 7 y 'l/Jx(m) = L J 1,(x - y) (my - ef;y(m))2 ·(3.2) 7 y Then define the sets f = { x 11 lef>x(m)I - m*(B)I > (m*({3) or 'l/Jx(m) > ((m*(/3))2} (3.3) where m*({3) is the largest solution of the equation x = tanh{3x, that is the location of the non- negative minimum of the function f /3. We recall (see e.g [E]) that m * ({3) = 0 if {3 ::; 1, m * ({3) > O if {3 > 1, lim/3too m* ({3) = 1 and lim13i1 (-;(~~W = 1. To simplify notation we will write m* = m* ({3) in the sequel. (, .2 < 1 will be chosen in a suitable way later. Note that if the boundary conditions are such that say ef>x(m(77)) +m*, then, if the configuration near the origin is such that c/Jo(m(u) < 0, :=::::: there must be a region enclosing the origin on which cjJ takes the value zero and thus belongs to f. For a reason that will become clear later, in a first step we will regularize this set. For this we introduce a second blocking of the lattice, this time on the scale of the range of the interaction. The points u of this lattice are identified with the blocks = { u x E ::zdj Ix - u/(1l)I ::; 1/(21£)} (3.4) just as in (1.4). We write in a natural way u(x) for the label of the unique block that c~:mtains x. We will call sets that are unions of such blocks u regular sets. We put = ro {xlu(x)nf,t:0} (3.5) For some positive integer n 2:: 1 to be chosen later, we now set (3.6) + where dist is the metric induced by the sup-norm on !Rd. n will depend on {3 and diverge as {3 1. The precise value of n will be specified later in (3.48). Notice that this definition assures that the set £ is a regular set in the sense defined above.. Connected components of the set set £ together with the specification of the values of mx, x E £ are called contours and are denoted by I'. For ar, such a contour, we introduce the notion of its boundary in the following sense: ={ ar x Er I dist(x, re) ::; n('Yl)-1} (3.7) 3 For that reason it is not possible to directly use the methods developed in [DZ] for studying low temperature phases of short range continuous spin models, although some of the ideas in that paper are used in our proof. 7 Note that by our definition of r we are assured that ar n r_ = 0. We denote by 0 = re n± {x I 14>x(m) =1= m*I ~Cm*} n (3.8) and call these regions ±-correct. Each connected component of the boundary of r connects either to n+ or n-. We will denote such connected components by art and 8I'i, respectively. r. r. For a connected set we denote by int the simply connected set obtained by "filling up the r.. r. holes" of This set is called the interior of a contour. The boundary of int will be referred to r.. ar r. as the exterior boundary of The connected component of that is also the boundary of int will be called exterior boundary of r and denoted by arext. The strategy to prove Theorem 1 is the usual one. First we observe that if boundary conditions are strongly plus, then in order to have that, say, lef>o(m) - m*I > (m*, it must be true that there exists a contour r such that 0 E int r. . Thus it suffices to prove that the probability of contours is sufficiently small. This will require a lower bound on the energy of any configuration compatible r., with the existence of and an upper bound on a carefully chosen reference configuration in which the contour is absent. We will show later (Lemma 3.8) that a lower bound on the energy can easily r.. be given in terms of the functions 4> and 'lj;, a fact that motivates the definition of The long range nature and of the interaction and the fact that the mx are essentially continuous yariables require the construction of the extensive "safety belts" around this set in order to assure an effective decoupling of the core of a contour from its exterior. The crucial reason for the definition of contours through the nonlocal functions ef> and 'ljJ is however the fact that these are "slowly varying" functions f. of x for any configuration m. Therefore, even if the core is very "thin" (e.g. a single point), one can show that on a much larger set Ie f>o (m ) - m *I or 'l/;x (m ) must still be quite large (e.g. half f). of what is asked for in This guarantees that in spite of the very thick "safety belts" we must construct around f, the energy of a contour compares nicely with its volume lr.I · We wiil now establish the "decoupling" properties. For this we must establish some properties ar of the configuration m on that minimizes Ear for given boundary conditions. Defi.nition 3.1: A configuration m'{;t is called optimal if m0Pt minimizes Ev(mv, mvc) for a given configuration mvc. f, An important point is that away from due to our definition of contours configurations must be close to constant in the following sense: Lemma 3.2: Assume that dist(x,f) > 1/("(£). Then {i) L J11.(x -y) (my± m*)2 ~ 4(2(m*)2 (3.9) y 8